The Los Cabos Lectures - BCCPbccp.berkeley.edu/beach_program/presentations09/CATBwhite1.pdf · EPS statistics for the standard ΛCDM cosmology Millennium Simulation cosmology: Ω

Dark Matter Halos

Simon WhiteMax Planck Institute for Astrophysics

The Los Cabos Lectures

January 2009

● Is all dark matter part of some halo?

● Was this always the case?

● How do halos grow? accretion? merging?

● How are they distributed?

● What is their internal structure? -- density profile -- shape -- subhalo population – mass/radial distributions, evolution -- caustics

● How do these properties affect DM detection experiments?

● How can they be used to test the standard paradigm?

● How do they affect/are they affected by the baryonic matter

Dark matter halos are the basic units of nonlinear structure

2.5cm100 kpc/h

CDM halos

A rich galaxy cluster halo Springel et al 2001

A 'Milky Way' halo Power et al 2002

A simple model for structure formation

In linear theory in a dust universe δ(x, z) = D(z) δ

o(x) = (2π)-3/2 D(z) ∫ d3k δ

k exp(-i k.x)

where we define D(0) = 1

Consider the smoothed density field δ

s(x, z; k

c) = (2π)-3/2 D(z) ∫

|k|<kc

d3k δk exp(-i k.x)

and define ‹δs(x, z; k

c) 2›

x = D2(z) σ

o

2(kc), M

s = 6π2ρ

ok

c

-3

As kc grows from 0 to ∞, the smoothing mass decreases from ∞ to 0,

and δs(x, z; k

c) executes a random walk

For a gaussian linear overdensity field Δδ

s = δ

s(x, z; k

c + Δk

c) - δ

s(x, z; k

c)

is independent of δs and has variance D2 Δσ

o

2

---- A Markov random walk ----

The “Press &Schechter” Ansatz

A uniform spherical “top hat” perturbation virialises when itsextrapolated linear overdensity is δ

c ≈ 1.69

Assume that at redshift z, the mass element initially at x is partof a virialised object with the largest mass M for which δ

s(x, z; k

c(M)) ≥ δ

c

This is the Markov walk's first upcrossing of the barrier δs = δ

c

The fraction of all points with first upcrossing below kc is then

the fraction of cosmic mass in objects with mass above Ms(k

c)

n(M, z) dM = exp - ( )2- ρo δ

c d ln σ

o

2 1 δc

√(2π) M2 D σo d ln M 2 D σ

o

Overdensityvs smoothing

at a given position

init

ial o

verd

ensi

ty

variance of smoothed fieldmassspatial scale

If the density field is smoothed using a sharp filter in k- space, then each step in the random walk is independent of all earlier steps

A Markov process

The walks shown at positions A and B are equally probable

A

B


at a given position

init

ial o

verd

ensi

ty


A

B

τ1

At an early time τ1

A is part of a quite massive halo

B is part of a very low mass halo orno halo at all

MA(τ

1) M

B(τ

1)?M

A(τ

1)


at a given position

init

ial o

verd

ensi

ty


A

B

τ2

Later, at time τ2

A's halo has grown slightly by accretion

B is now part of a moderately massive halo

MA(τ

2) M

B(τ

2)


at a given position

init

ial o

verd

ensi

ty


A

B

τ3

A bit later, time τ3

A's halo has grown further by accretion

B's halo has merged again and is now more massive than A's halo

MA(τ

3)M

B(τ

3)


at a given position

init

ial o

verd

ensi

ty


A

B

τ4

Still later, e.g. τ4

A and B are part ofhalos which follow identical merging/ accretion histories

On scale X they are embedded in a high density region.On larger scale Y in a low density region

XY

MA(τ

4)

MB(τ

4)

Consequences of the Markov nature of EPS walks

● The assembly history of a halo is independent of its future

● The assembly history of a halo is independent of its environment

● The internal structure of a halo is independent of its environment

● The mass distribution of progenitors of a halo of given M and z is obtained simply by changing the origin to σ

o

2(M) and δc/D(z)

● The resulting formulae can be used to obtain descendant distributions and merger rates

● A similar argument gives formulae for the clustering bias of halos

Does it work point by point?

Halo mass predicted for each particle by its own sharp k-space random walk

Mass of the halo in which the particle is actually found

Does it work statistically?

P&S74

Warren

Boylan-Kolchin et al 2009

Evolution of halo abundance in ΛCDM

Mo & White 2002

● Abundance of rich cluster halos drops rapidly with z

● Abundance of Milky Way mass halos drops by less than a factor of 10 to z=5

● 109M⊙ halos are almost as

common at z=10 as at z=0


Mo & White 2002 ● Temperature increases with both mass and redshift T ∝ M2/3 (1 + z)

● Halos with virial temperature T = 107 K are as abundant at z = 2 as at z=0

● Halos with virial temperature T = 106 K are as abundant at z = 8 as at z=0

● Halos of mass >107.5M⊙

have

T > 104 K at z=20 and so can cool by H line emission

8

10

1214


Mo & White 2002 ● Half of all mass is in halos more massive than 1010M

⊙

at z=0, but only 10% at z=5, 1% at z=9 and 10-6 at z=20

●1% of all mass is in halos more massive than 1015M

⊙

at z=0

●40% of all mass at z=0 is in halos which cannot confine photoionised gas

●1% of all mass at z=15 is in halos hot enough to cool by H line emission

8

10

1214


Mo & White 2002

● Halos with the abundance of L

* galaxies at z=0 are equally

strongly clustered at all z < 20

● Halos of given mass or virial temperature are more clustered at higher z

8

10

1214


Mo & White 2002

● The remnants (stars and heavy elements) from all star-forming systems at z>6 are today more clustered than L

* galaxies

● The remnants of objects which at any z > 2 had an abundance similar to that of present-day L

* galaxies are today more

clustered than L* galaxies

8

10

1214

Does halo clustering depend on formation history?

Gao, Springel & White 2005

The 20% of halos with the lowest formation redshifts in a 30 Mpc/h thick slice

Mhalo

~ 1011M⊙


The 20% of halos with the highest formation redshifts in a 30 Mpc/h thick slice

Mhalo

~ 1011M⊙

Does halo clustering depend on formation history?

Halo bias as a function of mass and formation time


Mhalo

= 1011M⊙/h

● Bias increases smoothly with formation redshift

● The dependence on formation redshift is strongest at low mass

● This dependence is consistent neither with excursion set theory nor with HOD modelsM

* = 6×1012M

⊙/h

Bias as a function of ν and formation time

Gao & White 2007

high zform

low zform

Bias as a function of ν and concentration

Gao & White 2007

high c

low c

Bias as a function of ν and main halo mass fraction

Gao & White 2007

high Fmain

low Fmain

Bias as a function of ν and subhalo mass fraction

Gao & White 2007

low Fsubhalo

high Fsubhalo

Bias as a function of ν and spin

Gao & White 2007

high spin

low spin

Halo assembly bias: conclusions

The large-scale bias of halo clustering relative to the dark matter depends on halo mass through ν = δ

c / D(z) σ

o(M) and also on

-- formation time -- concentration -- substructure content -- spin

The dependences on these assembly variables are different andcannot be derived from each other, e.g. more concentrated halosare more strongly clustered at low mass but less strongly clustered at high mass; rapidly spinning halos are more strongly clustered by equal amounts at all masses.

These dependences are likely to be reflected in galaxy bias

EPS statistics for the standard ΛCDM cosmology

Millennium Simulation cosmology: Ωm = 0.25, Ω

Λ = 0.75, n=1, σ

8 = 0.9

Angulo et al 2009

The linear power spectrum in “power per octave” form

Assumes a 100GeV wimpfollowing Green et al (2004)

free-streaming cut-off



Λ = 0.75, n=1, σ

8 = 0.9

Angulo et al 2009

Variance of linear density fluctuation within spheres containing mass M, extrapolated to z = 0

As M → 0, S(M) → 720

free-streaming cut-off



Λ = 0.75, n=1, σ

8 = 0.9

If these Markov randomwalks are scaled so themaximum variance is 720and the vertical axis is multiplied by √720, thenthey represent halo assembly histories for random CDM particles.

An ensemble of walks thusrepresents the probability distribution of assembly histories



Λ = 0.75, n=1, σ

8 = 0.9

Angulo et al 2009

Distribution of the masses ofthe first halos for a random set of dark matter particles

The median is 10-2M⊙

For 10% of the mass the first halo has M > 107M

⊙



Λ = 0.75, n=1, σ

8 = 0.9

Angulo et al 2009

Distribution of the collapse redshifts of the first halos for a random set of dark matter particles

The median is z = 13

For 10% of the mass the first halo collapses at z > 34

For 1% at z > 55



Λ = 0.75, n=1, σ

8 = 0.9

Angulo et al 2009

Distribution of the collapse redshifts of the first halos for dark matter particles split bythe mass of the first object

The high redshift tail isentirely due to matter in small first halos

For first halo masses belowa solar mass, the mediancollapse redshift is z = 21



Λ = 0.75, n=1, σ

8 = 0.9

Angulo et al 2009Total mass fraction in halos

At z = 0 about 5% (Sph) or 20% (Ell) of the mass is still diffuse

Beyond z = 50 almost all the mass is diffuse

Only at z < 2 (Sph) or z<0.5(Ell) is most mass in halos with M > 108M

⊙ The “Ell”

curve agrees with simulations



Λ = 0.75, n=1, σ

8 = 0.9

Angulo et al 2009

The typical mass element ina “Milky Way” halo goes through ~5 “infall events” where its halo falls into a halo bigger than itself.

Typically only one of theseis as part of a halo withM > 108M

⊙



Λ = 0.75, n=1, σ

8 = 0.9

Angulo et al 2009

The typical mass element ina “Milky Way” halo goes through ~3 “major mergers” where the two halos are within a factor of 3 in mass

The majority of these occur when the element is part of the larger halo

EPS halo assembly: conclusions

● The typical first halo is much more massive than the free streaming mass

● First halos typically collapse quite late z ~ 15

● Halo growth occurs mainly by accretion of much smaller halos

● There are rather few “generations” of accretion/merger events

● Major mergers are not a major part of the growth of many halos

Documents

The Los Cabos Lectures - BCCPbccp.berkeley.edu/beach_program/presentations09/CATBwhite1.pdf · EPS statistics for the standard ΛCDM cosmology Millennium Simulation cosmology: Ω